ADAMS OPERATORS AND FIXED POINTS

Transcription

1 ADAMS OPERATORS AND FIXED POINTS S. P. NOVIKOV Abstract. The aim of this article is to calclate the Conner Floyd invariants of the fixed points of the action of a cyclic grop, by analogy with Adams operators. We shall correct the mistakes made in a previos article. First of all, let me state that Appendices 3 and 4 in my article [2] contain fndamental errors. In particlar, an internal contradiction in the formlas of Theorem 3 in Appendix 4 was pointed ot to me by G. G. Kasparov, a stdent of the mechanico-mathematics faclty of Moscow State University. I am gratefl to him. The main aim of this article is to calclate definitively the Conner Floyd fnctions α 2n 1 (x 1,..., x n ) or fixed points by means of Adams operators in cobordism theory, introdced by the athor in [2] 1. I. Correction of the errors in Appendix 3 in the article [2] In Appendix 3 in the article [2], Lemma 1 and the constrction of the cell complexes S X in the general homology theory of X are incorrect. The basic Theorem 1 of this appendix is correct, since it follows from the Adams spectral seqence in the cobordism theory (this was shown in a footnote on p. 945). Here, by definition we mst have U (X, A) = U (X), k (Z), K (X), H (X) for A = Ω U, Z[x], Z[x, x 1 ], Z and U (X, A) = U (X), k (X), K (X), H (X) for the same Ω-modles A. This theorem follows easily from the identities and U (MU) Ω A = U (MU, A) Hom Ω (U (MU), A) = U (MU, A), which redce the second term of the Adams spectral seqence to the form Tor Ω (U (X), A) or Ext Ω (U (X), A). Theorem 2 is correct if we replace k by k (Z p ), since here the constrction of the cell complex is correct. In the remaining cases the constrction of the cell complexes is also correct, for example S k (BZ p ) and S U (BZ p ). Therefore all the applications of the reslts in Date: Received JUNE 14, Mathematics Sbject Classification. UDC Noted added in proof. Theorem 1 (see section III) was simltaneosly and independently obtained by A. S. Miščenko and G. G. Kasparov (in press). 1

2 2 S. P. NOVIKOV Appendix 3 to the classical lenses in Appendix 4 are correct (bt other mistakes occr there). II. Correction of the errors in Appendix 4 in the article [2] In Appendix 4 two problems were solved: 1. A complete calclation of the bordisms and cobordisms of the space BZ p by means of Adams operators in U-theory, and also a description of the strctre of the atomorphism λ x : BZ p BZ p, which is generated by mltiplication by an invertible reside x Z p (Theorems 1 and 2). Here there are no errors. 2. Calclation of the symmetric fnctions α 2n 1 (x,..., x n ) U 2n 1 (BZ p ) on a set of invertible resides x 1,..., x n, defined by the action of the grop Z p on the sphere S 2n 1, linear in the complex space C n and given by a diagonal matrix (a ij ), a jj = e 2πixj/p. We formlated three obvios reqirements that these fnctions mst satisfy: a) νa 2n 1 (x 1,..., x n ) = x 1 1 x 1 n a 2n 1, a 2n 1 H 2n 1 (BZ p ) is a standard basis element and ν : U H is a homomorphism of the agmentation ( normalization condition ). b) (λ x ) α 2n 1 (1,..., 1) = α 2n 1 (x 1,..., x 1 ), λ x was described above and (λ x ) : U 2n 1 U 2n 1 ( vales on the diagonal ). c) If two relations of the following form hold: γ (i) t α 2(k i) 1 (x (i) 1,t,..., x(i) k i,t ) = 0, i,t j,s δ s (j) α 2(l j) 1 (y (j) 1,s,..., y(j) l j,s ) = 0 (γ (i) t Ω 2i U, δ (i) s Ω 2j U ), then their direct prodct is also a relation of the form i,j,s,t γ (i) t δ s (j) α 2(k+l i j) 1 (x (i) 1,t,..., x(i) k i,t, y(j) 1,s,..., y(j) l j,s ) = 0 ( mltiplicative condition ). We also showed that from these three conditions we can recrsively calclate the vales of the fnctions α 2n 1 (x 1,..., x n ) in terms of α 2n 1 (1,..., 1), and we gave the reslt of the calclation for n = 2, 3 (Theorem 3). This theorem is not tre. The error lies in the fact that on page 946 (bottom) the elements α 2n 1 (1,..., 1) are incorrectly identified with the elements v 2n 1, which are canonically conjgate to the basis (v k ) in the complex S U (BZ p ). In fact (v 2n 1, v n 1 ) = 1 and (v 2n 1, v k ) = 0 for k n 1. In fact, we have the formlae (α 2n 1 (1,..., 1), v n 1 k ) = [CP k ] Ω U, α 2n 1 (1,..., 1) = v 2n 1 + [CP 1 ]v 2n [CP n 1 ]v 1. (I)

3 ADAMS OPERATORS AND FIXED POINTS 3 Also, on page 946 we wrote x instead of x 1 throghot, and in Lemma 3 we mst write (λ x ) v 5 = xv 5 + x2 x 3 [CP 1 ]v 3, 2 changing the sign of the second term. A correct recalclation leads to the following formlae: a) α 1 (x 1 ) = xα 1 (1); b) α 3 (x 1, y 1 ) = σ 2 α 3 (1, 1) + (σ 1 2σ 2 ) [CP 1 ] α 1 (1); 2 c) α 5 (x 1, y 1, z 1 ) = σ 3 α 5 (1, 1, 1) + (σ 2 3σ 3 ) [CP 1 ] α 3 (1, 1) 2 + (σ 1 2σ 2 + 3σ 3 ) [CP 1 ] 2 α 1 (1) + σ2 2 2σ 1 σ 3 3σ 3 ( 3 [CP 2 ] 4 σ 3 3 [CP 1 ] 2 4 ) α 1 (1), σ 1 are the elementary symmetric fnctions. In general, we have the following simple Lemma 1. The normalization, mltiplicative, and vale-on-the-diagonal conditions formlated above completely determine the fnctions α 2n 1 (x 1,..., x n ) for all p n. For the proof of the lemma we first consider primes p > n. By the normalization conditions we have α 2n 1 (x 1 1,..., x 1 n ) σ n α 2n 1 (1,..., 1) = i>0 (II) γ (n) i (x 1 1,..., x 1 n )α 2(n i) 1 (1,..., 1). If we know the fnctions α 2k 1 for all k < n, then, by virte of the mltiplicative condition, we can se the relations written above to form varios relations for α 2n 1 for all possible x 1,..., x n. These are linear eqations that are non-homogeneos, since we know α 2n 2 (x 1,..., x 1 ). It is easy to see that this system of eqations redces to trianglar form. First we find α 2n 1 (x 1, 1,..., 1), and so forth. This implies Lemma 1. For the remaining p the proof is similar, bt the linear eqations will not be over a field. Remark. It is possible in principle to find varios relations on α 2n 1 (x 1,..., x n ), by constrcting a large nmber of examples of actions of Z p on complex manifolds, and then to calclate α 2n 1 recrsively. Sch a method of calclation was realized by G. G. Kasparov for n = 2, 3, and he obtained the formla (II) in a different manner from that sed here. His method reqires the essential se of certain constrctions and theorems of Conner and Floyd. Kasparov first observed (from geometrical considerations) that the coefficients γ (n) i in α 2n 1 (x 1, 1,..., 1) = xα 2n 1 (1, 1,..., 1) + i>0 γ(n) i. α 2(n i) 1 (1,..., 1) depend only on i for i < n. This fact will be proved by algebraic methods and will be sed later.

4 4 S. P. NOVIKOV III. The complete calclation of the fnctions α 2n 1 (x 1,..., x n ) We shall seek the fnctions α 2n 1 (x 1,..., x n ) U 2n 1 (BZ p ) in the form α 2n 1 (x 1,..., x n ) = 1 α 2n 1 (1,..., 1) + γ (n) i (x 1,..., x n )α 2(n i) 1 (1,..., 1), σ n i>0 γ (n) i (x 1,..., x n ) Ω U, assming by definition that α 2(n i) 1 = 0 for i n. We denote by U 2 (BZ p ) the canonical two-dimensional cobordism class. Clearly we have k α 2n 1 (1,..., 1) = α 2(n k) 1 (1,..., 1), is the Čech excision. We denote by L 2n 1 BZ p the (2n 1)-dimensional skeleton formed by the classical lens with weights 1,..., 1, that represents the class α 2n 1 (1,..., 1). The Poincaré Atiyah dality operator for L 2n 1 is denoted by D n : U (L 2n 1 ) U (L 2n 1 ). We observe that U k (L 2n 1 ) = U k (BZ p ) for k < 2n 1. Therefore we can assme that α 2(n k) 1 (x 1,..., x n ) U (L 2n 1 ) for k > 0. We have the formlae D n+k α 2n 1 (1,..., 1) = k, D n+k α 2n 1 (x 1,..., x n ) = k B n (; x 1,..., x n ), B n (; x 1,..., x n ) does not depend on k. Here n+k = 0. By definition B n (; x 1,..., x n ) = σ 1 n + i 1 γ (n) i i = 1 k D n+kα 2n 1 (x 1,..., x n ). (III) Since for B n (; x 1,..., x n ) only the first n terms in the power series in make sense, we can consider B n (; x 1,..., x n ) to be an infinite series and at the end take n = 0. Or problem is to find a seqence of series B 1 (; x) = 1 x +..., B 2(; x, y) = 1 xy +..., B 3 (; x, y, z) = 1 xyz +... and so on, in calclating α 2n 1 (x 1,..., x n ) we consider only the first n coefficients. Namely, if then 1 B n (; x 1,..., x n ) = + x 1 x n i 1 γ (n) i i, α 2n 1 (x 1,..., x n ) = 1 n 1 α 2n 1 (1,..., 1) + γ (n) i α 2(n i) 1 (1,..., 1), σ n since α 2n 2i 1 (1,..., 1) = 0 for i n. The following important reslt holds.

5 ADAMS OPERATORS AND FIXED POINTS 5 Lemma 2. The seqence of series B 1, B 2,... is mltiplicative in the sense that B(; x) = B n (; x, 1,..., 1). B n (; x 1,..., x n ) = n B(; x i ), Proof. We seek fnctions α 2n 1 (x 1,..., x n ) as if Lemma 2 is tre. Having fond a soltion (see Theorem 1 below) we see whether these fnctions satisfy the normalization and mltiplicative conditions (the diagonal vales will be sed in determining the soltion, so they will atomatically be consistent). From the eqations (VIII), which follow from Theorem 1 (see below), it is clear that all of the reqirements are satisfied. Therefore Lemma 2 follows atomatically by virte of Lemma 1. Ths the problem redces to finding the series B(, x) = lim n B n(; x, 1,..., 1). For this prpose we se the atomorphism λ x : BZ p BZ p, generated by mltiplication by x Z p. Recall that in cobordism theory the Adams operator Ψ x = Ψ x U is given by the series Ψ x () = x 1 g 1 (xg()), g() = 0 [CP n ] n + 1 n+1 (the Miščenko series ) and g 1 is the inverse series. By definition we have λ x() = xψ x () = i 0 φ i (x) i+1, φ i (x) Ω U. We now se the general identity f (f (a) b) = a f (b) (IV) for arbitrary transformations f, a U, and b U. In this identity we sbstitte f = λ x, a =, b = α 2n+1 (1,..., 1) and observe that f α 2k 1 (1,..., 1) = α 2k 1 (x 1,..., x 1 ) for all k. Then from (IV) we obtain the eqation (λ x ) (xψ x () α 2n+1 (1,..., 1)) = i 0 φ i (x)α 2n 2i 1 (x 1,..., x 1 ) = α 2n+1 (x 1,..., x 1 ) = x n+1 α 2n 1 (1,..., 1) + γ (n+1) i (x 1,..., x 1 )α 2n 2i 1 (1,..., 1). (V) i

6 6 S. P. NOVIKOV When we apply the operator D n to this eqation, we find (bearing in mind that n = 0 and n ): n φ i (x)b n i (; x 1 ) i = B n+1 (; x 1 ), (V ) i 0 B n i (; x 1 ) i = D n α 2n 2i 1 (x 1,..., x 1 ). We divide the eqation (V ) by B n+1 (, x 1 )/. Then we obtain ( ) xψ x B(, x 1 =, (VI) ) and Therefore xψ x (v) = g 1 (xg(v)) = i 0 g(v) = n 0 and the series B(; x) is of the form B(; x) = [CP n ] n + 1 vn+1. = x 1 Ψ x 1 ()B(, x 1 ) φ i (x)v i+1 xψ x () ; B n(; x 1,..., x n ) = n x i Ψ xi (). (VII) Since α 2n 1 (x 1,..., x n ) = B n (; x 1,..., x n ) α 2n 1 (1,..., 1), (VII) implies the following definitive reslt. Theorem 1. We have the formla α 2n 1 (x 1,..., x n ) = xψ x () = g 1 (xg()), x i Ψ xi () α 2n 1(1,..., 1), g = n 0 [CP n ] n + 1 n+1 (observe that by definition a k α 2n+1 (1,..., 1) = aα 2n 2k+1 (1,..., 1), a Ω U is an arbitrary scalar ). If the grop Z p acts in a complex manner on the manifold M n with fixed points P 1,..., P q, at which it has a collection of weights x (j) 1,..., x(j) n Z p, j = 1,..., q, then we have the eqations 2 q n = 0, (VIII) j=1 x (j) i Ψ x(j) i () 2 These eqations will be referred to as the Conner Floyd eqations.

7 ADAMS OPERATORS AND FIXED POINTS 7 is a formal variable that generates the ring Ω U [] with relations pψ p () = 0 and n+1 = 0. We take p to be a prime. III. Implications of the Conner Floyd eqations For transformations with isolated fixed points (q of them) the eqations (VIII) are actally S(n) nmerical eqations mod p, at least for large primes p > n, involving the qn variables x (j) i Z p, x (j) i 0 mod p; here S(n) is the nmber of symmetric polynomials in n variables of degree n (or S(n) = n k=0 rk Ω2k U ). We observe that the collections of weights {x (j) i } of the action of Z p are geometrically indistingishable if we perform an arbitrary permtation of the indices i and the indices j and mltiply the whole collection {x (j) i } by a nmber λ 0 mod p. In fact the eqations (VIII) are defined on projective space P qn 1 over Z p with the hypersrfaces P ij [x (j) i = 0] eliminated. Therefore the manifold of types of action of Z p lies in P qn 1 \ ij P ij, factored by the prodct of the permtation grops S n S q (see [2]). From (II) we can already draw certain conclsions: 1. For n = 2 and p = 3 the nmber q of necessary fixed points of the action of Z 3 cannot be two. For n = 2 and p 5 we can have an action of Z 3 with two fixed points. 2. For n = 3, nlike the case of n = 2, we can have an action of Z p with two fixed points for p For n = 3 and q = 2 (two points), of the for eqations ( (II) (since S(3) ) = 4) is a conseqence of the other two, since the coefficient of [CP 2 ] 3 [CP 1 ] 2 4 α 1 (1) is a homogeneos fnction of σ 1, σ 2, σ 3 of degree 1. Therefore it changes sign when the σ i change sign. For the two points P 1 and P 2 the Conner Floyd eqations (II) and (VIII) reqire a change of sign. This fact is an accidental coincidence. 4. Since for arbitrary q the nmber of eqations S(n) grows mch faster than qn (namely, S(n) 1 2π eπ 2n 3 2n (1 + O(n 1 4 ))), for each nmber q there exists an n = n(q) sch that for N > n(q) we know that every complex action of Z p on N-dimensional complex manifolds has more than q fixed points. However, a rigoros proof reqires a demonstration of the independence of approximately S(n) of the eqations (VIII), at least for large p. In view of possible coincidences, as in the preceding sbsection, this reqires an additional algebraic analysis of the eqations. 5. We consider the real case corresponding to a change from U -bordism to SO - bordism. Here we restrict or attention to grops of odd order. The corresponding fnctions will be denoted by α R 2n 1(x 1,..., x n ) SO 2n 1 (BZ p ), α R 2n 1 = r α 2n 1, r : U SO is the realizing homomorphism. Clearly the fnctions possess the following symmetry: α R 2n 1( x 1,..., x n ) = α R 2n 1(x 1,..., x n ),

8 8 S. P. NOVIKOV corresponding to rotations of the sphere S 2n. The Conner Floyd eqations (VIII R ) are obtained from (VIII) by means of the realizing operator r. These are actally S(k) nmerical eqations for dimensions n = 4k + 2 and n = 4k + 4, since n 1 S(k) = rk Ω i SO Z p. i=0 If there is no general interdependence between these eqations apart from the symmetry indicated above, then, as in sbsection 5, this implies the following reslt: a) If the nmber of points q is odd, then there exists a dimension n(q) sch that on manifolds of dimension N > n(q) there is no action of Z p with q isolated fixed points. b) If q = 2t, then for dimensions N > n(q) every action of Z p is sch that the points P 1,..., P q split into pairs P 1, P 1,..., P t, P t for which the collections of weights of the points P j and P j are opposite in sign. Here p is odd and the manifolds are even-dimensional and orientable. The dimension n(q) can be calclated if we know the nmber of independent eqations in (VIII R ). If this nmber increases faster than linearly, then the reslt is tre. If it increases more slowly than pn, then n(q) =. Apparently this nmber increases as S(n/4), bt I have not been able to prove it; ( n ) S 4 1 4π 2n eπ/2 2n 3. IV. Nmerical realizations of the Conner Floyd eqations How can we find these eqations most effectively? Here Ψ x = x 1 g 1 (xg()), g = [CP n ] n + 1 n+1. Since CP n is algebraically independent in Ω U, then g is practically a series with arbitrary coefficients. Bt the series x iψ x i () is of the form i 0 γ(n) i i, and sometimes it is possible to calclate effectively some of the Chern nmbers of the coefficients γ (n) i, which gives varios nmerical realizations of the eqations (VIII). We introdce the following important examples of sch realizations: a) T -gens (corresponding to the homomorphism U K ). Since T [CP n ] = 1 and T (M N) = T (M)T (N), g() goes into We have and g T () = n+1 = ln(1 ). n + 1 g 1 T () = 1 e T (xψ x ()) = 1 (1 ) x.

9 ADAMS OPERATORS AND FIXED POINTS 9 Finally we obtain the eqation j i 1 (1 ) x(j) i n+1 = 0, k (1 (1 ) p ) = 0, k 0. = 0, (VIII T ) b) L-gens (signatre). Since L[CP 2n ] = 1, L[CP 2n+1 ] = 0, g goes into g L = 2n+1 2n + 1 = 1 ( ) ln 1 and Therefore We have the eqation g 1 L = th. ( ( )) x 1 + g 1 L (xg L()) = th 2 ln = (1 + )x (1 ) x 1 (1 + ) x + (1 ) x. j i (j) (1 + )xi (1 + ) x(j) i + (1 ) x(j) i (1 ) x(j) i, (VIII L ) n+1 = 0, k (1 + )p (1 ) p (1 + ) p = 0, k 0. + (1 ) p c) The Eler characteristic c (of the tangent bndle). Since c n [CP n ] = n+1, it follows that g c () = 1 and g 1 c = + 1. Therefore gc 1 x (xg c ()) = 1 + x. We have the eqation 1 + x (j) i = 0, (VIII j i x (j) c ) i n+1 = 0, k p = 0, k p d) The t-characteristic, which selects polynomial generators in Ω U, t k = c ω, ω = (k). We observe that t(m i N j ) = 0 for i > 0 and j > 0. Since t[cp n ] = n + 1, it follows that g t () = 1, (g 1 ) t = v v2 1 v. ( 1 [g 1 (xg())] t = x 1 x ) 1 x

10 10 S. P. NOVIKOV and [ ] g 1 = 1 (xg()) t x [ n ] x i Ψ x = i () Finally we have the eqation j t i ( 1 n (1 1 + ( x 1 x ), ) x i 1 x i (n 1). x 1 x n x(j) i 1 x (j) i x (j) 1 x (j) n n+1 = 0, k [pψ p ()] t = 0, k 0. IV. Q-genera of general form ) (n 1), (VIII t ) We obtained for types of nmerical eqations. It is possible to dedce a general nmerical eqation corresponding to a mltiplicative Hirzebrch seqence of Chern classes Q(z) = 1 + a 1 z + a 2 z Of special interest are the A-genera A = Q(z) = z/2 sh z/2 and Ā = 2z sh 2z. Ā = ec1/2 T. We can only se an A-gens for odd p in view of the lack of spinors. For an arbitrary mltiplicative seqence W (z) we have a Q-gens Q[M n ], and we mst calclate g Q = Q[CP n ] n + 1 n+1, g 1 Q, g 1 Q (xg Q) and /g 1 Q (xg Q). n 0 Since Q[CP n ] is the component of z n in the series Q n+1 (z), it follows that Q[CP n ] = 1 Q n+1 (z) 2πi z n+1 dz. z =ε Hence we obtain dg Q d = Q[CP n ] n = 1 2πi n 0 n 0 = 1 Q(z)/z, dz = 1 2πi 2πi Therefore 1 Q(z) z g Q () = 1 2πi 0 z =ε z =ε dz z/q(z), ( ) n+1 Q(z) dz z Q z < 1. dz d z/q(z) for small. Integrating with respect to, we find that g Q () = 1 ( ln 1 Q(z) 2πi z z =ε, < z Q(z) ) dz = ( ) 1 z (). Q(z)

11 Let φ(z) = we have that ADAMS OPERATORS AND FIXED POINTS 11 z Q(z) and let φ 1 (v) be the inverse fnction. From the form of g Q () g Q () = φ 1 (), g 1 Q (v) = v Q(v). Therefore g 1 Q (xg Q()) = xg Q Q(xg Q ). Conseqently, the general Conner Floyd eqation for a Q-gens takes the form j i Q(x(j) i g Q ()) g Q () x (j) i = 0, n+1 = 0, k pg Q () = 0, k 0, Q(pg Q ()) g Q () = n 0 Q[CP n ] n + 1 n+1, g Q () Q(g Q ()) =. (VIII Q ) We consider the case Q = c = 1 1+z, Q = A and Q = Ā. The fnctions are every to be nderstood as formal series in, (1 + y) α = 1 + αy + α(α 1) y e) The normal Eler characteristic c. Since Q = 1 1+z, it follows that Therefore We have the eqation f) A-genera: g Q () = 1 2 ( ). g 1 Q (xg Q) = x 4 ( )(2 + x( )). j i 4 n+1 = 0, (2 + x(j) i ( )) 1 x (j) i ( = 0. (VIII c ) ) k g 1 Q (pg Q()) = 0, k 0. Q = A = z/2 sh z/2, Q = Ā = 2z sh 2z. For an A-gens we obtain ( ) g A () = 2 ln , 4 g 1 A () = Q() = 2 sh 2.

12 12 S. P. NOVIKOV Frthermore = g 1 A (xg A) = 2 sh ( x ln ( 2 + ( ) x ( )) ) x. Finally we have the following eqation for odd p: ( ) x ( ) x = 0. j i n+1 = 0, k g 1 A (pg A()) = 0, k 0. (VIII A ) By analogy we write down the eqations (VIII c ), (VIII T ), (VIII L ), (VIII c ), (VIII t ), (VIII A ), (VIII Ā ) for the case in which we have an action of the grop Z p for which all fixed manifolds have a trivial normal bndle. V. Global invariants of the manifold carrying the action of Z p Here we consider the problem of calclating the characteristic nmbers and integral Q-genera modlo p of the manifold M n that carries the action of Z p. For simplicity we limit or attention, as before, to the case of isolated fixed points P 1,..., P q of the whole grop as the only singlarities. Let the weights (for the point P j ) be x (j) 1,..., x(j) n. This problem was solved for a nmber of cases by Conner and Floyd. However, de to its simplicity, we shall indicate its soltion here. In fact, the answer follows easily from an old work of Tamra [1] in which he constrcted classes (Pontrjagin, Chern) of p-manifolds. In addition, I may remark that if, in the case considered by Tamra, the action withot fixed points can be extended from the bondary M onto M, then all the Tamra characteristic nmbers are eqal to zero mod p. We consider the Tamra p-manifold M n (x 1,..., x n ) = D 2n, Z p ( D 2n ), Z p acts linearly on S 2n 1 with weights x 1,..., x n. Tamra calclated his classes for M 2 (x 1, x 2 ), writing down an awkward answer. The general answer for the Chern Tamra nmber c ω [M n (x 1,..., x n )] is of the form c ω = v ω(x 1,..., x n ) x 1 x n, dim ω = n and v ω = x a1 i 1 x an i n is a symmetrized ω-monomial. Hence we obtain in a simple way a formla for the action of Z p on M n : q v ω (x (j) 1,..., x(j) n ) = c x (j) 1 x (j) ω [M n ] (mod p), (IX) n j=1 For large primes p > n + 1, this formla also correctly describes T -genera, L- genera and A-genera. For primes p n + 1, the formla is incorrect for these

13 ADAMS OPERATORS AND FIXED POINTS 13 Q-genera. For example the formla T = [ n j x (j) i 1 e x(j) i contains p in the denominator for p n+1. Ths it mst be interpreted as follows: if the x (j) i are integers sch that x (j) i (mod p) = x (j) i, then we have the formla [ n ] 1 x (j) i = T [M n ] (mod p), (IX j σ n (j) T ) 1 e x(j) i n (mod p) the divisibility by p of the expression in parentheses is a conseqence of eqations (VIII) for the weights x (j) i of the fixed points. In the holomorphic and real cases, the formlae (IX), (IX T ), etc., can apparently be derived from the reslts of Atiyah and Bott [3] and some methods from nmber theory. For the L-gens, the A-gens and any other Q-gens the proof is similar. ] n 1 σ (j) n VI. The action of a circle with fixed points We shall consider complex actions of the circle S 1 on a manifold M n, having only isolated fixed points for the whole grop (for sbgrops there may also be other singlarities). At each of these fixed points P the action of S 1 on the tangent space T P is given by a diagonal matrix A P (φ), the a ij (φ) = e ixjφ, φ S 1, x j are integers. We let π denote the collection of all primes that divide the collection {x j }, and we let Ẑπ denote the completion of the integers in the topology in which the open sets are the ideals generated by all the nmbers that are relatively prime to the collection π. Let G π = Char Ẑπ be the grop of characters. It is easy to see that the representation A P (φ) determines a qasi-complex action withot fixed points of the grop G π on the sphere S 2n 1, enclosing the singlar point P, with weights x j, the action of a character h G π = Char Z π is given by the action of the element h(1) S 1. This action is determined by the collection of integers {x j }, which may depend on the collection of primes π. By definition, the grop of complex bordisms BG for the grop G is the collection of pairs (M, G) of qasi-complex manifolds with the corresponding action of G withot fixed points, identifying pairs with respect to cobordism. We call dim U dim G the dimension of a pair. Sppose that the action of S 1 on the manifold M n has fixed points P 1,..., P q with weights x (j) i, i = 1,..., n, j = 1,..., q. Then we take π to be a collection from all the primes that divide the set x (j) i and define Ẑπ analogosly. We set G π = Char Z π. The linear representation A Pj (φ) at the point P j on the sphere S 2n 1 determines an element α(x (j) 1,..., x(j) n ) U (BG π ).

14 14 S. P. NOVIKOV Clearly we have the ( Conner Floyd ) relation α(x (j) 1,..., x(j) n ) = 0, P j G π = Char Ẑπ. We have the following simple Lemma 3. For the natral homomorphism G π S 1 the indced mapping is an isomorphism. Later we shall consider the fnctions ρ: U (BG π ) Ẑπ U (CP ) Ẑπ ρα(x (j) 1,..., x(j) n ) U (CP ) Ẑπ and denote them by β(x (j) 1,..., x(j) n ), x (j) 1,..., x(j) n are integers that are invertible in Ẑπ. How can we calclate the fnctions β(x 1,..., x n )? If U 2 (CP ) is a canonical element and the α = CP i U 2i (CP ) are standard bordisms, then we have the following obvios formlae: a) j α i = α i j ; b) β(1,..., 1) = α i 1 ; c) β ( 1 x,..., x) 1 = [λx ] β(1,..., 1), λ x : CP CP is generated by mltiplication by x in the grop S 1. d) λ x(η) = η x, η K(CP ) is a canonical one-dimensional bndle; e) λ x() = xψ x (x), U 2 (CP ) is a canonical element. As for the case Z p, so also here we seek a series B(; x) sch that β(x 1,..., x n ) = n B(, x i ) α n 1, B(; x) = 1 x By repeating the argments employed in the proof of Theorem 1, we finally obtain the following assertion. Theorem 1a. We have the formla β(x 1,..., x n ) = n the x i are invertible elements in Ẑπ, and x i Ψ xi () α n 1, β(x 1,..., x n ) U 2n 2 (CP ) Ẑπ, α i = [CP i ]. Let s make a few remarks abot other grops. For a tors the investigation is completely similar. Also, it is easy to carry it ot for compact (connected) commtative grops.

15 ADAMS OPERATORS AND FIXED POINTS 15 VII. Arbitrary finite grops It is known (Zassenhas) that grops G that can act discretely and orthogonally on spheres have cyclic Sylow p-sbgrops for p > 2 and highly special 2-sbgrops ( generalized qaternions ). Therefore their homologies can be analyzed very simply. Let G be sch a grop and let { 1,..., m } be its complete set of nitary irredcible representations, acting discretely on spheres. Then all representations sch that = k j j, k j 0 also have this property. We denote this semigrop by R + G. The following fnctions are defined on R+ G : a( ) U 2n 1 (BG), R + G, n = dim c. The fnction of the Eler class in cobordism theory (or the older Chern class), σ n ( ) U 2n (BG), K(BG), is also defined. We observe that for sms of one-dimensional fibers η = η i we have n σ n (η) = σ 1 (η i ), and if η i = η xi, then, by definition, n σ n (η) = σ 1 (η xi ) = n x i Ψ xi (σ 1 (η)). Theorem 1b. We have the following general formla: dim c 1 = dim c 2 = n, i R + G. α( 2 ) = σ n( 1 ) σ n ( 2 ) α( 1) U 2n 1 (BG), For commtative grops we proved this formla earlier (cf. Theorem 1), since for the canonical representation η of the grop Z p we have σ 1 (η) = U 2 (BG) and σ 1 (η x ) = xψ x (). In the general case the proof can be redced to Theorem 1 by the sal argments involving restrictions to the Sylow p-sbgrops. Since for all p > 2 they are cyclic, the formla (X) will have been dedced from Theorem 1 if we tensor mltiply it by all the rings of p-adic integers for p > 2. For p = 2, we mst analyze the 2-grops, i.e., the grops of generalized qaternions G t 2, t 2: a, b G t 2, a 2t = b 4 = 1, b 2 = a 2t 1, aba 1 = b 1. In view of the simple strctre of these 2-grops, the proof can be carried ot by direct calclations, and we complete it below in examples 1 and 3. Remark. The first coefficient σ n ( 1 )/σ n ( 2 ) = γ( 1, 2 ) +..., γ is a scalar, determines the degree of the eqivariant sphere mapping S 2n 1 S 2n 1 with respect to the actions of 1, 2. Example 1. Let G 2 = G 2 2 be the grop of qaternions (of order eight) with generators a and b and relations a 2 = b 2 = (ab) 2 (center), a 4 = b 4 = 1 and aba 1 = b 1. There exists a niqe irredcible representation R + G, given (X)

16 16 S. P. NOVIKOV by the Pali matrices and acting discretely on S 3. The ring of representation has a basis 1,, a, b, c R G2, dim = 2, dim a = dim b = dim c = 1, and a 2 = b 2 = c 2 = 1, ab = c, 2 = (1 + a)(1 + b), a = b =. There are defined elements a 4k 1 (k ) U 4k 1 (BG 2 ) and the dal elements w j, ω = σ 2 ( ). The element a G 2 generates the grop Z 4, and when restricted to Z 4, the representation becomes ρ + ρ 1, ρ is the basis representation (e 2πi/4 ). The order of the element σ 2 (ρ + ρ 1 ) in the grop U 4 (L 4k+3 ) eqals 2 2k+1, becase of previos reslts abot Z 4, L 4k+3 is the lens of dimension 4k + 3 over Z 4. Clearly σ 2 (ρ + ρ 1 ) = Dα 4k 1 (1, 1,..., 1, 1). From the properties of the ring R G it is easy to conclde that the orders of the elements σ 1 ( ) = a j w j (a j Ω U ), σ 2 ( ) = w do not exceed 2 k+1. Hence 2 2k w 0 and 2 2k+1 w = 0. Therefore the restriction of this cyclic grop to U (L 4k+3 ) is a monomorphism. We mst consider the elements α(k ) U 4k 1 (BG 2 ). On the basis of the above, their orders eqal 2 2k+1, and w j α(k ) = α((k j) ). Since the restriction of the elements w i to Z 4 eqals σ i 2(ρ + ρ 1 ) = [ 2 Ψ 1 ()Ψ 1 ()] i, from the eqation 4Ψ 4 () = 0 it is easy to derive relations in the Ω-modle between the w j for varios j, and apply this to the action of G 2 on M n, having the manifolds M i of fixed points of varios dimensions with trivial normal bndles (i.e., of the form [M j ] α((n j) ). Example 2. Let G be a grop of order 120, acting discretely and nitarily on S 3, G/[G, G] = 1 and 1 Z 2 G S 5 + 1, S+ 5 is the grop of even permtations of five elements. The Sylow sbgrops of G are G 2, Z 3, and Z 5, and the corresponding Weyl grops W p (G) of inner atomorphisms of G that preserve the Sylow p-sbgrops G 2, Z 3, and Z 5 are of the form: W 2 = At G = S 3, W 3 = W 5 = Z 2, W 3 (G) and W 5 (G) act by the atomorphism x x 1 on Z 3 and Z 5. We observe that the restrictions of cohomologies, K-theory, and U-theory to Sylow sbgrops are invariant with respect to W p (G). There are two irredcible nitary representations 1, 2 R G that act discretely on spheres (namely, on S 3 ). Their restrictions to G 2, Z 3, and Z 5, respectively, are of the form 1 [ (G 2 ), x + x 1 (Z 3 ), t + t 1 (Z 5 )], 2 [ (G 2 ), x + x 1 (Z 3 ), t 2 + t 2 (Z 5 )] (XI) x = (e 2πi/3 ), t = (e 2πi/5 ) are basis representations. There exists an oter atomorphism : G G sch that 1 = 2. We observe that H (BG) = Z 120 [y], dim y = 4. The following Conner Floyd elements are feasible: α(k 1 + l 2 ) U 4k+4k 1 (BG). By virte of the strctre of the ring H (BG), the collection of elements α(n 1 ) U 4n 1 is a complete basis of U (BG) as an Ω-modle. Let w = σ 2 ( 1 ). Clearly w j α(n 1 ) = α((n j) ).

17 ADAMS OPERATORS AND FIXED POINTS 17 From the formla (XI) and Theorem 1b we conclde that α 4k 1 ((k l) 1 + l 2 ) = w l σ 2 ( 2 ) l α(k 1). this formla may be restricted to Z 5, since 1 and 2 coincide when restricted to G 2 and Z 3. After restriction to Z 5 we obtain ( a) (w/σ 2 ( 2 )) l Ψ 1 l () 4Ψ 2 ()Ψ ()) U (BZ 2 5 ), b) w/σ 2 ( 2 ) = 1 + γ, γ has order 5 h(n) in any n-dimensional skeleton of BG; on sch a sbgrop the restriction to BZ 5 is an isomorphism. When we write U (BG) as the sm of its p-adic parts for p = 2, 3, 5, then, in principle, we can se the preceding formlas to carry ot all the necessary ring calclations and draw conclsions abot the fixed points. Example 3. Let G = G t 2, t > 2 be the grop of generalized qaternions with generators a, b (a 4 = b 2t = 1, bab 1 = a 1 and b 2t 1 = a 2 ). Since G/[G, G] = Z 2 Z 2t 1 with generators a and b, the grop H has 2 t irredcible one-dimensional representations 1, η, ρ, ρ 2,..., ρ 2t 1, ηρ,..., ηρ j, η(a) = 1, η(b) = 1 and ρ(a) = 1, ρ(b) = e 2πi 2 t 1. Also, the grop G has 2 t 2 irredcible representations 1,..., 2t 2 of dimension two, acting discretely on S 3. Namely, j (a) = iσ x ; j (b) = iα j σ y, σ x, α y, σ z are the Pali matrices, and the α j are nmbers sch that αj 2t 1 = 1 for t > 2, σx 2 = σy 2 = 1. There exist in all 2 t 2 distinct roots α j of 1 to within sign (α j, α j ), determining the representations j, j 2 t 2. The restriction of the representation j to the cyclic grop Z2 t = (b) prodces an element p 2j 1 + p 1 2j, α j is the (2j 1)-th power of a primitive root of 1 of degree 2 t. The general form of the elements is α( k j j ) U (BG), the initial elements may be taken as the collection α(k 1 ) U 4k 1 (BG); all the remaining elements are expressed in terms of these with coefficients in Ω U. Similarly we introdce the elements w k = σ2 k ( 1 ) that are dal to the elements α(l 1 ). It can be shown that on the Ω U -modle Ω[w] U (BG) the restriction to the cyclic sbgrop Z2 t = (b) is a monomorphism. This follows from the Atiyah identity K(BG) = RĜ and the Conner Floyd homomorphism σ 1 : K 0 U 2 by analogy with Example 1. Ths, the order of the element α 4k 1 (2x 1 1, 1 2x 1,..., 2x k 1, 1 2x k ) U 4k 1 (BZ 2 t) eqals precisely 2 t+2(k 1), which is also the order of α( xj ). Therefore all the calclations connected with Ω U [w] can be performed after restriction to U (BZ 2 t), and w j becomes (2j 1) 2 Ψ 2j 1 ()Ψ 1 2j (), = σ 1 (ρ) U 2 (BZ 2 t). Examples 1 and 3 prove the formla (X) of Theorem 1b, since it was proved earlier for cyclic grops (in Theorem 1).

18 18 S. P. NOVIKOV A final remark on the actions and possible singlar orbits of these actions for finite grops G on complex manifolds (we assme that all singlar orbits are isolated): we mst consider stationary sbgrops of orbits G P G at their fixed points P sch that G P acts on the tangent sphere S 2n 1 at the point P withot fixed points (and G P is maximal at the point P), defining the invariants α P U 2n 1 (BG P ). Then it is easy to obtain the general Conner Floyd eqations for the action of the grop G on M n with isolated singlarities. By what we have shown above, it sffices to consider only maximal cyclic sbgrops G P G P in the final calclations (cf. Examples 1 and 3). From the class of conjgate points P i with respect to G/G P we mst select one. The imbedding G P G means that the homomorphisms ρ : U (BG P ) U (GB) are defined. The general Conner Floyd eqations are of the form: ρ (α P ) = 0 U (BG), (XII) G P for each G P (to within a transformation) we mst in (XII) select one point from the collection g(p), g G/G P. VIII. Another application of Adams operators in cobordism theory Theorem 2. The Thom complex M(ξ) of an arbitrary element ξ K(BU (n) ) is homotopically eqivalent to M U with preservation of the homological Thom isomorphism if and only if the difference ξ η is of the form ξ η = a i k Ni i (Ψ k i 1)ξ i, η is a niversal bndle, BU (n) is a finite skeleton of BU, and Ψ k is the sal Adams operator in K-theory. This theorem nfortnately does not imply the well-known Adams hypothesis. Therefore we shall not give a complete proof. Roghly speaking, the proof is based on the fact that the operators Ψ 1/k in U -theory are atomorphisms of MU in the category S Z(1/k), sch that Ψ 1/k /H 2(N+i) (MU n ) is mltiplication by k i. On the other hand, for a niversal element η K(BU) there exists a transformation ρ k : M(Ψ k η) MU, which, in the homologies H 2(N+i) is mltiplication by k i. Therefore the morphism Ψ 1/k ρ k : M(Ψ k η) MU commtes with the Thom isomorphism. Bibliography [1] I. Tamra, Characteristic classes of M-spaces. I, J. Math. Soc. Japan 11 (1959), MR 22 #5046. [2] S. P. Novikov, Methods of algebraic topology from the point of view of cobordism theory, Izv. Akad. Nak SSSR Ser. Mat. 31 (1967) = Math. USSR Izv. 1 (1967), MR 36 #4561. [3] M. F. Atiyah and R. Bott, Notes on the Lifschitz fixed point theorem for elliptic complexes, Harvard Univ., 1964, preprint; Rssian transl., Matematika 10 (1966), no. 4,